MQUP eBooks· 2020· Book chapter
Conjectures and Refutations
- 3,391citations
- 2020year
Short summary
Conjectures are not merely failures to prove theorems but an essential and rational part of mathematical research, driving progress through trial and error.
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Key points
- Conjectures are a vital and rational component of mathematical research, not a sign of weakness.
- The method of conjecture and refutation is a core driver of progress in mathematics.
- Examples from linear systems, algebraic varieties, and tensor ranks demonstrate the role of conjectures.
- This perspective challenges the traditional view of mathematics as solely a system of definitive truths.
AI-generated from the title and abstract; the full text is not read.
Abstract
To the outside observer, mathematics may appear as the definite system of truth. From this vantage point conjectures may seem a fringe phenomenon to be shunned; the poor man's theorem and an admittance of one's unsuccessful attempts of proving it. I will argue that aforementioned conjecture is false. I will argue for Karl Popper's view that "there is no more rational procedure than the method of trial and error--of conjecture and refutation: of boldly proposing theories; of trying our best to show that these are erroneous; and of accepting them tentatively if our critical efforts are unsuccessful." We will see that conjectures are an integral and essential part of research in the mathematical sciences. This will be illustrated with some surprising examples of conjectures and their refutations concerning the complexity of solving linear systems, the dimensions of algebraic varieties, and the equality of symmetric and nonsymmetric rank of symmetric tensors.
The authors' abstract, as published at the source. MQUP eBooks, 2020 · DOI ↗
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Field: Computational Theory and Mathematics
Computational Theory and MathematicsComputer Science